Let $F_1 = (0,1)$ and $F_ 2= (4,1).$  Then the set of points $P$ such that
\[PF_1 + PF_2 = 6\]form an ellipse.  The equation of this ellipse can be written as
\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.\]Find $h + k + a + b.$
Explanation: We have that $2a = 6,$ so $a = 3.$  The distance between the foci is $2c = 4,$ so $c = 2.$  Hence, $b = \sqrt{a^2 - c^2} = \sqrt{5}.$

The center of the ellipse is the midpoint of $\overline{F_1 F_2},$ which is $(2,1).$  Thus, the equation of the ellipse is
\[\frac{(x - 2)^2}{3^2} + \frac{(y - 1)^2}{(\sqrt{5})^2} = 1.\]Hence, $h + k + a + b = 2 + 1 + 3 + \sqrt{5} = \boxed{6 + \sqrt{5}}.$